# 庞加莱不等式

## 叙述

### 经典形式

p是一个大于等于1实数n是一个正整数。${\displaystyle \Omega }$n欧几里得空间${\displaystyle \mathbb {R} ^{n}}$上的一个有界子集，并且其边界是满足利普希兹条件的区域（也就是说它的边界是一个利普希茨连续函数的图像）。在这种情况下，存在一个只与${\displaystyle \Omega }$p有关的常数C，使得对索伯列夫空间${\displaystyle \mathbb {W} ^{1,p}(\Omega )}$ 中所有的函数u，都有：

${\displaystyle \|u-u_{\Omega }\|_{L^{p}(\Omega )}\leq C\|\nabla u\|_{L^{p}(\Omega )}}$

${\displaystyle u_{\Omega }={\frac {1}{|\Omega |}}\int _{\Omega }u(y)\,\mathrm {d} y}$

### 推广

${\displaystyle [u]_{H^{1/2}(\mathbf {T} ^{2})}^{2}=\sum _{k\in \mathbf {Z} ^{2}}|k|{\big |}{\hat {u}}(k){\big |}^{2}<+\infty :}$

${\displaystyle \int _{\mathbf {T} ^{2}}|u(x)|^{2}\,\mathrm {d} x\leq C\left(1+{\frac {1}{\mathrm {cap} (E\times \{0\})}}\right)[u]_{H^{1/2}(\mathbf {T} ^{2})}^{2},}$

## 参考来源

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3. ^ M, Bebendorf, A Note on the Poincar´e Inequality for Convex Domains (PDF), Journal for Analysis and its Applications, 2003, 22 (4): 751–756, （原始内容 (PDF)存档于2012-05-26）
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